1. Ch. 6 – The Definite Integral
6.5 – Trapezoidal Rule

2. Cut into 6ths, then make trapezoids using the L and R endpoints!
Now that we learned how to find exact area under a curve using integrals, let’s estimate area with trapezoids? Boooo!
Recall: area of a trapezoid =
Ex: Estimate the area of the region bounded by the graph of f(x) = 9 – x2 and the x- and y-axes using the trapezoidal rule and 6 subintervals.
x = 3
Cut into 6ths, then make trapezoids using the L and R endpoints!

3. Ex (cont’d): Note that this estimate is equal to
f(x) 9
0.5
8.75 1 8
1.5
6.75 2 5
2.5
2.75 3
Ex (cont’d):
Note that this estimate is equal to
the average of LRAM6 & RRAM6!
Is this estimate an overestimate
or an underestimate?
It’s an underestimate because
the curve is concave down!
x = 3

4. Ex (cont’d): The actual area of f(x) = 9 – x2 over [0, 3] is 18
Ex (cont’d): The actual area of f(x) = 9 – x2 over [0, 3] is 18. Use that answer to find the percent error in the trapezoidal rule estimate found on the last slide.
So the trapezoidal sum underestimated the area by 0.694%.

5. Ex: The temperature in my classroom from 6am to 6pm is shown below
Ex: The temperature in my classroom from 6am to 6pm is shown below. Use the Trapezoidal Rule to approximate the average temperature for the 12 hour period.
Average temperature = average value!
Since we don’t have a function to integrate, let’s use trapezoid rule to approximate the area underneath the curve!
Time (h after 6am) 1 2 3 4 5 6 7 8 9 10 11 12
Temp (°F) 54 60 71 70 68 65 64 63 66 67 69 72

6. Simpson’s Rule = To approximate
Simpson’s Rule = To approximate , partition [a, b] into an even # (n) of subintervals of length h = (b-a)/n:
Use the program on your calculator! Show what h and n are for your work!
Ex: Use Simpson’s Rule to approximate the area underneath y = 9 – x2 using 6 subintervals over [0, 3].
n = 6, h = ½ …